Jeffrey Schlupp, Andros Townsend and Luka Milivojevic were the goal scorers for Crystal Palace.
Leicester City's Premier League match away to Cardiff on Saturday will go ahead as planned.
Luke Shaw sealed the points seven minutes from time before Jamie Vardy headed home a consolation for the visitors.
In six seasons at Leicester, Vardy, 31, has scored 88 goals and most memorably spearheaded the Foxes' incredible Premier League triumph in 2015/16.
England are scheduled to lock horns with Sweden at Samara Arena in the third quarterfinal of the tournament on Saturday.
Joe Hart and Jack Wilshere fail to make the cut in the FIFA World Cup squad.
Jamie Vardy scored the goal that effectively ensured Leicester City would remain in the Premier League as the outgoing champions won 1-0 away to West Bromwich Albion.
Jamie Vardy scored in vain as Leicester City's Champions League fairytale came to a valiant end following a 1-1 quarter-final draw at home to Atletico Madrid on Tuesday.
England have 13 points, four ahead of Slovakia with just the nine group winners in Europe guaranteed a place at the finals in Russia in 2018 World Cup.
Jamie Vardy's first-half opener was Leicester's first league goal in seven games and made them the last team from Europe's five major championships to find the net in 2017.
At a rain-sodden King Power Stadium, Jamie Vardy hit his first top-flight treble and Andy King also found the net to ensure Leicester City atoned for their 5-0 Champions League mauling by Porto.
Manchester United, who have confirmed they are close to re-signing former player Paul Pogba from Juventus, open their Premier League campaign at Bournemouth next Sunday
Several football stars like Olivier Giroud, Antonie Griezmann, Zlatan Ibrahimovic hogged the limelight during Euro 2016 for their tattoos as well
Jamie Vardy, presently on duty with England at the Euro 2016 finals, has agreed to an improved four year contract offer from Leicester City, spurning Arsenal's advances after they activated a
Slovakia must now wait to see if they qualify as one of four best third-place best teams after their goalkeeper Matus Kozacik pulled off a string of save.
England hope to extend their perfect record versus Slovakia. Live streaming of England vs Slovakia Group B Euro 2016 match will be available on all digital platforms like web and mobile. Catch live action through NDTV live blog too
England manager Hodgson has an embarrassment of riches up front, but Harry Kane, who started in the 1-1 draw against Russia and win over Wales, has struggled to make an impact in France
Daniel Sturridge scored a dramatic late goal after Jamie Vardy had cancelled out Gareth Bale's free-kick opener to help England beat Wales 2-1